Vector Space Definition
A vector space consists of a set of V ( elements of V are called vectors), a field F ( elements of F are scalars) and the two operations
 Vector addition is an operation that takes two vectors u, v âˆˆ V, and it produces the third vector u + v âˆˆ V
 Scalar Multiplication is an operation that takes a scalar c âˆˆ F and a vector v âˆˆ V and it produces a new vector uv âˆˆ V.
Where both the operations must satisfy the following condition
Elements of V are mostly called vectors and the elements of F are mostly scalars. There are different types of vectors. To qualify the vector space V, the addition and multiplication operation must stick to the number of requirements called axioms. The axioms generalise the properties of vectors introduced in the field F. If it is over the real numbers R is called a real vector space and over the complex numbers, C is called the complex vector space.
Vector Space Axioms
All the axioms should be universally quantified. For vector addition and scalar multiplication, it should obey some of the axioms. Here eight axiom rules are given.
Conditions for Vector Addition
An operation vector addition â€˜ + â€˜ must satisfy the following conditions:
Closure : If x and y are any vectors in the vector space V, then x + y belongs to V
 Commutative Law : For all vectors x and y in V, then x + y = y + x
 Associative Law : For all vectors x, yand z in V, then x + ( y + z ) = ( x + y ) + z
 Additive Identity : For any vector x in V, the vector space contains the additive identity element and it is denoted by â€˜ 0 â€˜ such that 0 + x = x and x + 0 = x
 Additive inverse : For each vector x in V, there is an additive inverse x to get a solution in V.
Condition for Scalar Multiplication
An operation scalar multiplication is defined between a scalar and a vector and it should satisfy the following condition :
Closure: If x is any vector and c is any real number in the vector space V, then x. c belongs to V
 Associative Law: For all real numbers c and d, and the vector x in V, then c. (d. v ) = ( c . d). v
 Distributive law: For all real numbers c and d, and the vector x in V, (c + d). v = c . v + c . d
 Distributive law: For all real numbers c and the vectors x and y in V, c. ( x + y) = c. x + c. y
 Unitary Law : For all vectors x in V, then 1. v = v . 1 = v
Vector Space Properties
Here are some basic properties that are derived from the axioms are

 The addition operation of a finite list of vectors v_{1} v_{2}, . . . , v_{k} can be calculated in any order, then the solution of the addition process will be the same.
 If x + y = 0, then the value should be y = âˆ’x.
 The negation of 0 is 0. This means that the value of âˆ’0 = 0.
 The negation or the negative value of the negation of a vector is the vector itself: âˆ’(âˆ’v) = v.
 If x + y = x, if and only if y = 0. Therefore, 0 is the only vector that behaves like 0.
 The product of any vector with zero times gives the zero vector. 0 x y = 0 for every vector in y.
 For every real number c, any scalar times of the zero vector is the zero vector. c0 = 0
 If the value cx= 0, then either c = 0 or x = 0. The product of a scalar and a vector is equal to when either scalar is 0 or a vector is 0.
 The scalar value âˆ’1 times a vector is the negation of the vector: (âˆ’1)x = âˆ’x. We define subtraction in terms of addition by defining x âˆ’ y as an abbreviation for x + (âˆ’y).
x âˆ’ y = x + (âˆ’y)
All the normal properties of subtraction follows
 x + y = z then the value x = z âˆ’ y.
 c(x âˆ’ y) = cx âˆ’ cy.
 (c âˆ’ d)x = cx âˆ’ dx
Vector Space Problems
Go through the vector space problem provided here.
Question :
Show that each of the conditions provided is in vector space
 The set of linear polynomials \(p_{1}=\left \{ a_{0}+a_{1}xa_{0},a_{1}\epsilon \mathbb{R} \right \}\) under the addition and scalar multiplication operations
 Under usual matrix operation, the set of 2 x 2 matrix with real entries
 Three component row vectors with usual operations
 The set A = \(\left \{ \begin{pmatrix} x\\ y\\ z\\ w\end{pmatrix}\epsilon \mathbb{R}^{4}x+yz+w=0 \right \}\) under the operations from\(\mathbb{R}^{4}\)
Solution :
Conditions checked from the axioms and properties:
 The zero element is 0 + 0x is zero
 The zero element of the vector space under 2 x 2 matrix is zero
 The zero element of three component row vectors are zeros
 The closure property of addition involves \(\begin{pmatrix} x_{1}\\ y_{1}\\ z_{1}\\ w_{1} \end{pmatrix}+\begin{pmatrix} x_{2}\\ y_{2}\\ z_{2}\\ w_{2} \end{pmatrix}=\begin{pmatrix} x_{1}+x_{2}\\ y_{1}+y_{2}\\ z_{1}+z_{2}\\ w_{1}+w_{2} \end{pmatrix}\) is in A because (x_{1}+x_{2}) + (y_{1}+y_{2}) – (z_{1}+z_{2}) + (w_{1}+w_{2}) = ( x_{1}+y_{1}z_{1}+w_{1})+(x_{2}+y_{2}z_{2}+w_{2}) = 0 + 0
Similarly, the closure of scalar multiplication can be obtained.
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